Abstract We study a Lax pair in 2-parameter Lie algebra various representations. The overlap coefficients of the eigenfunctions L and standard basis are given terms orthogonal polynomials functions. Eigenfunctions for operator $$\mathfrak {sl}(d+1,\mathbb {C})$$ sl ( d <mml:mo...

A bstract We formulate the stringy nonrelativistic limits of flat space and AdS 5 × S string as coset models, based on Bargmann extended Newton-Hooke algebras respectively. Our construction mimics typical relativistic one, but differs in several interesting ways. Using our formulation we give a Lax representation equations motion both models.

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere’s presentation of metric spaces and Barr’s presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in ...

We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for AN−1 by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic CalogeroMoser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connectio...

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseu...

Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Bäcklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings ar...

The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but...

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospect...